Convergence results and stability of some integral equations on time scale

The aim of this thesis is to examine the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of Volterra integral equations on time scales defined as an arbitrary, non-null and closed subset of the set of real numbers R. The thesis also examines the existence or uniqueness of the solutions of Fredholm integral equations given on time scales in different spaces. The thesis prepared for these purposes consists of six chapters. The first chapter is devoted to the introduction and brief information is given about the historical development of the integral equation, time scale and stability concepts. In the second part, the basic information, theorems and examples used in the thesis are presented in detail under the title of basic concepts. Other parts contain the original results of the thesis. In the third chapter, firstly, two different types of Volterra integral equations given on time scales are generalized and Ulam-Hyers and Ulam-Hyers-Rassias stabilities are examined for these equations. In addition, a non-linear Volterra integral equation is defined on the time scale, Ulam-Hyers and Ulam-Hyers-Rassias stability results for this equation are proven and an example supporting these results is presented. In the last part of this section, it is shown that a general Volterra integral equation given on the time scale has Ulam-Hyers-Rassias stability. In the fourth chapter, it is shown that a T-mean non-expanding transformation in b-metric-like spaces has a single fixed point, and the (T,S)-stability result of the T-Picard iteration method using this transformation is given. With the result of the fixed point theorem, the existence and uniqueness of the solutions of the nonlinear Fredholm-Hammerstein and linear Fredholm integral equations defined on time scales have been proven. Additionally, two numerical examples from different time scales are presented to support the findings. In the fifth chapter, first published in 2022 by Abbas et al. The AA-iteration method introduced by , in Banach space, has been modified to hyperbolic metric space. Then, weak w^2-stability results for contraction transformations in such spaces and strong convergence and Δ-convergence results for generalized (α,β)-non-expansive transformations are examined. In addition, a numerical example for generalized (α,β)-non-expandable transformations is presented and a comparative convergence analysis is performed with different iteration methods. Finally, the findings were evaluated with applications on the nonlinear Fredholm-Hammerstein and linear Fredholm integral equations on the time scale. The last section is completed with the conclusion and recommendations section. One of the articles produced from this thesis was published in the Maltepe Journal of Mathematics and the other two were published in the AIMS Mathematics journal. The mentioned articles will serve as a source for future studies on the stability of integral equations on the time scale and the existence or uniqueness of their solutions. Keywords: Time scale, integral equations, stability, fixed point, nonexpansive mappings